A dynamical polysystem consists of a family of continuous dynamical systems, all acting on a given metric space. The first chapter of the present thesis shows a generalization of control systems via dynamical polysystems and establishes the equivalence of the two notions under certain lipschitz condition on the function defining the dynamics. The remaining chapters are focused on a basic theory of dynamical polysystems. Some topological properties of limit sets are described in Chapter 2. Chapters 3 and 4 provide characterizations for various notions of strong stability. Chapter 5 makes use of the theory of closed relations to study Lyapunov functions. Prolongations and absolute stability make the object of the last chapter.